108 XIII INTEGRATION Let X be a compact space and let p be a real-valued function on #R(X) satisfying the following two conditions: (i) p(fjrg) ^p(f)Jrp(g)\ 00 p(af) = ap(f) for all a > 0. Then the set P of functions /e ^R(X) such that p(f) <; 0 is a convex cone. Suppose furthermore that (iii) inf/(*) ^p(f) ^sup/(x) for all /e^R(X), so that we have xeX xsX p(l) = i. Show that there exists a positive measure jit of mass 1 on X such that fi(/)^X/) for all/e ^R(X). If moreover we are given a linear form u on a vector sub- space E of ^R(X) such that u(f) ^p(f) for all /e E, then there exists a measure ju, of the above type which extends the form u. (Consider a denumerable total set (gn)nzo ^ ^R(X), with #o=l; use the result of Problem 1 inductively to obtain, on the subspace G of ^R(X) generated by E and the gn, a linear form v such that ~p(—f) ^v(f)^p(f) for all/e G, and deduce that v extends by continuity to a measure on X.) Show that the measure p, is unique if and only if p(f) -f p(—f) — 0 for all/e E. 4. THE VAGUE TOPOLOGY Since the space MC(X) is a subspace of C^**, we can define the weak topology, i.e., the topology of simple convergence in «?fc(X) (12.15). This topology on MC(X) is called the vague topology. To say that a sequence (^M) of measures on X converges vaguely to a measure p therefore means that, for each function /e Jf C(X), the sequence (/^n(/)) converges to (i(f) in C. (1 3.4.1 ) Let (jun) be a sequence of measures on X such that, for eachfe 3C C(X), the sequence (/*„(/)) tends to a limit u,(f) in C. Thenf\— >u.(f) is a measure on X and is the vague limit of the sequence (jun). If the fln are all positive, then so is ju. We have already remarked (13.1) that, for each compact subset K of X, jf(X; K) is a Banach space and the restrictions of the u.n to Jf* (X; K) are continuous linear forms. Hence it follows from the Banach-Steinhaus theorem (12.16.5) that the restriction of IJL to «^(X; K) is continuous, and therefore that /i is a measure on X (and clearly a positive measure if the \in are positive). We recall (12.15) that a subset H of MC(X) is said to be vaguely bounded (or just bounded, if there is no risk of ambiguity) if, for each /e Jf C(X), we have sup \u(f)\ < 4-oc. Every vaguely convergent sequence is vaguely /*eH bounded. (1 3.4.2) Let H be a bounded subset of MC(X). (i) For each compact subset K of X, there exists a real number CK > 0 such that, for each fi e H and eachfe «>f C(X), we have (and therefore (13.3.2.1), M(|/|) £ CK /has a finite derivative on the